Let and be real parameters

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

a\,(\neq 0)

and

b

be real parameters. Consider the integral involving an exponential function whose exponent is linear in

x

. (i) Show that

\displaystyle \int (ax+b)\,e^{(ax+b)}\,dx

can be written entirely in terms of

(ax+b)

and the constants

a

and

b

. Include the constant of integration. (ii) Using the expression obtained in part (i), evaluate

\displaystyle \int_{0}^{\ln 3} (2x+1)\,e^{(2x+1)}\,dx

. (iii) State and justify how your result in part (i) is affected, if at all, when

a

is replaced by

-a

while

b

remains unchanged.